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# How many integers less than 1000 have no factors (other than

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How many integers less than 1000 have no factors (other than [#permalink]  23 Aug 2010, 12:57
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How many integers less than 1000 have no factors (other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420
[Reveal] Spoiler: OA
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Re: good one [#permalink]  23 Aug 2010, 13:17
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mehdiov wrote:
How many integers less than 1000 have no factors(other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420

First of all it should be "how many positive integers less than 1000 have no factors (other than 1) in common with 1000", as if we consider negative integers answers will be: infinitely many.

$$1000=2^3*5 ^3$$ so basically we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5.

Multiples of 2 in the range 0-1000, not inclusive - $$\frac{998-2}{2}+1=499$$;
Multiples of 5 in the range 0-1000, not inclusive - $$\frac{995-5}{5}+1=199$$;
Multiples of both 2 and 5, so multiples of 10 - $$\frac{990-10}{10}+1=99$$.

Total # of positive integers less than 1000 is 999, so # integers which are not factors of 2 or 5 equals to $$999-(499+199-99)=400$$.

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Re: good one [#permalink]  03 Sep 2010, 04:24
Bunuel,

Yes -- we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5 = which done not have 2/5 as a factor.

For this we can USE the VENN diagram technique as shown below

The integers <= 1000 divigible by 2 = 1000/2 = 500, but = 499 if 1000 is excluded
The integers <= 1000 divigible by 5 = 1000/5 = 200, but = 199 if 1000 is excluded
The integers <= 1000 divigible by 10(2*5) = 1000/10 = 100, but = 99 if 1000 is excluded

hence, integers that r divisible by 2only and 5only = 500+200-100 (or 499+199-99 if 1000 excluded)= 600 (599 if 1000 is excluded)

so the answer is 1000-600 (or 999 - 599) = 400.

am i correct.
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Re: good one [#permalink]  03 Sep 2010, 04:40
Expert's post
muralimba wrote:
Bunuel,

Yes -- we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5 = which done not have 2/5 as a factor.

For this we can USE the VENN diagram technique as shown below

The integers <= 1000 divigible by 2 = 1000/2 = 500, but = 499 if 1000 is excluded
The integers <= 1000 divigible by 5 = 1000/5 = 200, but = 199 if 1000 is excluded
The integers <= 1000 divigible by 10(2*5) = 1000/10 = 100, but = 99 if 1000 is excluded

hence, integers that r divisible by 2only and 5only = 500+200-100 (or 499+199-99 if 1000 excluded)= 600 (599 if 1000 is excluded)

so the answer is 1000-600 (or 999 - 599) = 400.

am i correct.

Yes, it's correct. Basically the same way as used in my post.
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Re: good one [#permalink]  03 Sep 2010, 08:01
I agree the answers are basically the same
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Re: good one [#permalink]  11 Jan 2011, 01:55
Bunuel, why can't we simply divide 1000 by 2 to find the number of multiples of 2? My reasoning is that every second number is a multiple of 2 so there must be exactly 500 numbers.

Thanks.
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Re: good one [#permalink]  11 Jan 2011, 02:03
Expert's post
nonameee wrote:
Bunuel, why can't we simply divide 1000 by 2 to find the number of multiples of 2? My reasoning is that every second number is a multiple of 2 so there must be exactly 500 numbers.

Thanks.

There are 100/2=500 multiple of 2 in the range 1-1000 INCLUSIVE. As we need numbers LESS than 1000 which are also multiples of 2 then we should subtract 1 from that number. So there are total of 500-1=499 multiples of 2 in the range 0-1000, not inclusive.
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Re: good one [#permalink]  23 Jul 2011, 05:27
Hi!
I have a book with this question and it says, that the correct answer 401...i see that there is no such answers in your questions...so i really confused..can somebody explain why it can be 401? or it is a 100% mistake?
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Re: How many integers less than 1000 have no factors (other than [#permalink]  07 Jan 2013, 11:18
The question asks for the number of integers less than 1000 and other than 1.
Isnt one included in the 400 integers that you are claimimg to be the answer?
Answer should be 399 if we exclude 1.
Please correct me in case i missed something.
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Re: How many integers less than 1000 have no factors (other than [#permalink]  07 Jan 2013, 19:54
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rohantiwari wrote:
The question asks for the number of integers less than 1000 and other than 1.
Isnt one included in the 400 integers that you are claimimg to be the answer?
Answer should be 399 if we exclude 1.
Please correct me in case i missed something.

The question does not ask you to exclude 1.

Every positive integer less than 1000 has one common factor with 1000. What is it? It is 1.
1 is a common factor between any two positive integers.

If the question were: How many positive integers less than 1000 have no factors in common with 1000 ?
Then the answer would be 0. There are no positive integers which have no common factors with 1000. All the positive integers have a common factor and that is 1. But the question wants to know the number of positive integers which have no common factor other than 1 (1 will always be a common factor). Basically, it is looking for positive integers which are co-prime with 1000.
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Re: How many integers less than 1000 have no factors (other than [#permalink]  01 Apr 2013, 02:25
all odd numbers excluding odd multiples of 5 have only 1 as common factor with 1000.
hence 500 odd numbers-((995-5)/10)+1)= 400
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Re: How many integers less than 1000 have no factors (other than [#permalink]  02 Apr 2013, 11:30
consider integers between 1 and 100 - half of them are even - hence 50 integers are multiples of 2 ( which also includes even multiples of 5) + 10 odd multiples of 5 = 60
Hence 40 integers that are not multiples of 2 and/or 5 -
hence considering integers between 1 and 1000 - there are 40*10 = 400 integers which do not have common multiple with 1000 other than 1.
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Re: good one [#permalink]  03 Oct 2013, 13:09
Bunuel wrote:
mehdiov wrote:
How many integers less than 1000 have no factors(other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420

First of all it should be "how many positive integers less than 1000 have no factors (other than 1) in common with 1000", as if we consider negative integers answers will be: infinitely many.

$$1000=2^3*5 ^3$$ so basically we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5.

Multiples of 2 in the range 0-1000, not inclusive - $$\frac{998-2}{2}+1=499$$;
Multiples of 5 in the range 0-1000, not inclusive - $$\frac{995-5}{5}+1=199$$;
Multiples of both 2 and 5, so multiples of 10 - $$\frac{990-10}{10}+1=99$$.

Total # of positive integers less than 1000 is 999, so # integers which are not factors of 2 or 5 equals to $$999-(499+199-99)=400$$.

What about the prime numbers Bunuel ?? For ex : 7. Neither its a multiple of 2, nor 5 and it does not has any common factors with 1000 (except 1)
So, shouldn't the answer include prime numbers between 1-999 as well. And if YES, how do we calculate the number of primer numbers from 1-999 ???
Plz clarfily.

Thanks.
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Re: good one [#permalink]  04 Oct 2013, 00:01
Expert's post
sumitchawla wrote:
Bunuel wrote:
mehdiov wrote:
How many integers less than 1000 have no factors (other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420

First of all it should be "how many positive integers less than 1000 have no factors (other than 1) in common with 1000", as if we consider negative integers answers will be: infinitely many.

$$1000=2^3*5 ^3$$ so basically we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5.

Multiples of 2 in the range 0-1000, not inclusive - $$\frac{998-2}{2}+1=499$$;
Multiples of 5 in the range 0-1000, not inclusive - $$\frac{995-5}{5}+1=199$$;
Multiples of both 2 and 5, so multiples of 10 - $$\frac{990-10}{10}+1=99$$.

Total # of positive integers less than 1000 is 999, so # integers which are not factors of 2 or 5 equals to $$999-(499+199-99)=400$$.

What about the prime numbers Bunuel ?? For ex : 7. Neither its a multiple of 2, nor 5 and it does not has any common factors with 1000 (except 1)
So, shouldn't the answer include prime numbers between 1-999 as well. And if YES, how do we calculate the number of primer numbers from 1-999 ???
Plz clarfily.

Thanks.

We counted multiples of 2 or 5 in the range 0-1000, not inclusive and then subtracted that from total number of integers in the range 0-1000. The number we get contains all numbers which are not multiples of 2 or 5, thus all primes (apart from 2 and 5) in that range too.

Hope it's clear.
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Re: How many integers less than 1000 have no factors (other than [#permalink]  28 Oct 2014, 12:18
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Re: How many integers less than 1000 have no factors (other than   [#permalink] 28 Oct 2014, 12:18
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